Extracting key frames from long animation
Bake your animations


Make a backup copy
Select all the bones involved, then in Pose mode go to Pose > Animation > Bake Animation.
In the pop-up window:

Set Start Frame, End Frame, and Frame Step (in your example set it to 10).

Check "Only selected" (only selected bones will be baked); "Visual keying" (the position after all modifiers and constraints will be baked); "Overwrite current action" (you've made a backup copy, right??)

The "Clear constraints" option will remove all constraints from the armature after baking, making it perfectly animatable by the new baked action.

"Clear parents" involves object mode only, so it doesn't apply here.

Set "Bake data" to "pose"
Bake!